Step by step
The question
Question: which number has a z-score of −1.5, when the mean is 60 and the SD is 4?
z = −1.5: mean 60, SD 4. x?
A z-score of −1.5 means 1.5 SDs below the mean.
z = −1.5: 1.5 SDs below the mean
1 Write the formula
Start at the mean, then go z SDs: x is the mean plus z times the SD.
x = mean + z × SD
Careful!
A negative z-score means below the mean.
So x comes out smaller than the mean.
Careful: a negative z means below the mean: x is smaller
2 Put the numbers in
−1.5 times 4 is −6.
60 minus 6 is 54.
x = 60 + (−1.5) × 4 = 60 − 6 = 54
3 Read it
54 on the number line: 1.5 SDs below the mean.
4 Check
Let's check: the z-score of 54 is 54 minus 60, over 4.
That is −1.5.
So it is right!
z = 54 − 604 = −64 = −1.5
Let's look back.
First: write the formula.
Then: put the numbers in.
Then: read it.
Last: check.
Answer
x = 60 + (−1.5) × 4 = 60 − 6 = 54